Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
Can you work through these direct proofs, using our interactive proof sorters?
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Follow the hints and prove Pick's Theorem.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Can you find the areas of the trapezia in this sequence?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
Can you invert the logic to prove these statements?
Sort these mathematical propositions into a series of 8 correct statements.
Can you rearrange the cards to make a series of correct mathematical statements?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Have a go at being mathematically negative, by negating these statements.
This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
These proofs are wrong. Can you see why?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Here the diagram says it all. Can you find the diagram?
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Can you discover whether this is a fair game?
This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
Explore a number pattern which has the same symmetries in different bases.
Which of these roads will satisfy a Munchkin builder?
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
An introduction to some beautiful results of Number Theory